Nim, nim.py and

games.py

Homework 4 Problem 4

The History of Nim Games • Believed to have been created in China;

unknown date of origin • First actual recorded date- 15th century

Europe • Originally known as Tsyanshidzi meaning “picking stones game”

• Presently comes from German word “nimm” meaning “take”

Adapted from a presentation by Tim Larson and Danny Livarchik

Rules of Nim •  Impartial game of mathematical strategy • Strictly two players • Alternate turns removing any number of items

from any ONE heap until no pieces remain • Must remove at least one item per turn • Last player to be able to remove a wins • Variations:

– Initial number of heaps and items in each – Misere play: last player who can move loses – Limit on number of items that can be removed

Demonstration

Player 1 wins!

Theoretical Approach • Theorem developed by Charles Bouton in 1901 • This states that in order to win, the goal is to reach

a nim-sum of 0 after each turn until all turns are finished

• Nim Sum: evaluated by taking the exclusive-or of the corresponding numbers when the numbers are given in binary form

• Exclusive-or is used for adding two or more numbers in binary and it basically ignores all carries

Tree for (2,1)

Tree for (2,2)

games.py • Peter Norvig’s python framework for multiple-

player, turn taking games •  Implements minimax and alphabeta • For a new game, subclass the Game class

– Decide how to represent the “board” – Decide how to represent a move – A state is (minimally) a board and whose turn to move – Write methods to (1) initialize game instance, (2)

generate legal moves from a state, (3) make a move in state, (4) recognize terminal states (win, lose or draw), (5) compute utility of a state for a player, (5) display a state

Assumptions about states • games.py assumes that your representation

of a state is a object with at least two attributes: to_move and board

• The Struct class defined in utils.py can be used to create such instances – s = Struct(foo=‘a’, to_move = 1, board = [[1][2][3]]) – Access the attributes as s.to-move, etc.

Caution

• Python lists are mutable objects •  If you use a list to represent a board and whant to

generate a new board from it, you probably want to copy it fist new_board = board[:] new_board[3] = new_board[3] - 1

Players The games.py framework defines several players •  random_player: choses a random move from

among legal moves •  alphabeta_player: uses alpha_beta to choose

best move, optional args specify cutoff depth (default is 8) and some other variations

•  human_player: asks user to enter move

Variations def make_alphabeta_player(N): """ returns a player function that uses alpha_beta search to depth N """ return lambda game, state: alphabeta_search(state, game, d=N) # add to the PLAYER dictionary player function named ab1,ab2,...ab20 # that use alpha_beta search with depth cutoffs between 1 and 20 for i in range(20): PLAYER['ab'+str(i)] = make_alphabeta_player(i)